The sharp refined Bohr inequalities for a subclass of close-to-convex harmonic mappings

Abstract

Let $\mathcal{H}$ be the class of normalized complex valued harmonic functions $f=h+\overline{g}$ defined on the unit disk $\mathbb{D}$, where $h$ and $g$ are analytic functions with the normalization conditions $h(0)=h^{\prime }(0)-1=0$ and $g(0)=0$. For the class $R_H^0(\gamma ,\delta ,\lambda )$ ( $0\le \lambda <\gamma \le \delta$) consisting of functions \( f = h+\bar{g} \in \mathcal{H}\) satisfying the condition $f_{\overline{z}}(0)=0$ and the inequality $Re(\gamma h^{\prime }(z)+\delta z{h}^{\prime \prime }(z)+(\frac{\delta -\gamma }{2})z^2{h}^{\prime \prime \prime }(z)-\lambda )>|\gamma g^{\prime }(z)+\delta z{g}^{\prime \prime }(z)+(\frac{\delta -\gamma }{2})z^2{g}^{\prime \prime \prime }(z)|$, we obtain sharp improved Bohr Phenomenon, refined Bohr radius and the Bohr-Rogosinski inequality for the class $R_H^0(\gamma ,\delta ,\lambda )$.

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